UNIVERSITATIS MARIAE CURIE-SKLODOWSKA LUBLIN - POLONIA
VOL. LI. 1,3 SECTIO A 1997
SANDOR CSÓRGÓ (Ann Arbor and Szeged)
Renyi-Mixing in the Generalized Arc-Sine Law*
Dedicated to Professor Dominik Szynal on the occasion of his 60th birthday
Abstract. It is shown that the proportion of positive sums of independent and identically distributed random variables in the generalized arc-sine law is Renyi-mixing under Spitzer’s classical necessary and sufficient condition for the law. Some consequences for the number of positive sums in a random number of games are derived, along with an extension of the original law for Revesz-dependent sequences of random variables.
1. Introduction. Following Renyi [18] and Renyi and Revesz [20], we say that a sequence {£n}£Li of random variables, given on some probabil
ity space (fi,A,P), is mixing with the limiting distribution function G(-) if P{Rn < y} n A} -» G(t/)P{A} at every continuity point y of G on the real line R, for each event A £ A, where an unspecified convergence rela
tion is meant to hold as n —> oo throughout this note. To separate the notion clearly from many other types of mixing, for which the word is more customary, we shall refer to it as Renyi-mixing. Since G is assumed to
Work partially supported by the US National Science Foundation, Grant DMS-96- 25732.
be a proper non-degenerate distribution function, in general Renyi-mixing of {£n}^i with the limiting distribution function G demands more than convergence in distribution, contained for the choice of A = Q. In fact, according to Renyi’s [18] own characterization, given also in Sections 5.7 and 5.8 of his book [19], it happens if and only if P{£n < y} —* G(y) and P{Rn < y} n Rfc < J/}} —► G(y)P{£k < y} for every continuity point y of G, for each fixed ł £ N:= {1,2,...}.
Besides its intrinsic interest, one motivation for Renyi’s mixing exten
sion of a limit theorem is the preservation of that limit theorem under the change of the probability measure P to another probability measure Q < P, absolutely continuous with respect to P. It also turned out that Renyi- mixing in a limit theorem is the proper framework in which that same limit theorem may be transferred to extended versions for randomly selected sub
sequences of the original underlying sequence of random variables; see for example work by the author [4] and more recently by Rychlik [22] and nu
merous references therein. Both types of consequences will be discussed below for the number of positive sums that we consider in the present note.
Let A\,X2,... be a sequence of independent, identically distributed ran
dom variables .given on some probability space (Q,A,P), and let F(i) :=
P { A < x}, x £ R, denote the common distribution function. Set Sj :=
+ • • • + Xj, j € N, for the partial sums and, for any event A £ A, let /(A) denote its indicator, defined for all w € Q as JW(A) = 1 if tv G A and JW(A) = 0 if u> A. Then 52j=i > 0), the occupation time of the positive half-line by the first n sums, is often thought of as the number of times a gambler with cumulative gains Si,... , Sn, which may very well be losses, is ahead of his opponent in n games. Initiated by Levy [16]**
and subsequently enriched in contributions by Erdos and Kac [11], Chung and Feller (particularly for the exact discrete arc-sine law for coin toss
ing described by Feller [12] including his own combinatorial improvements) and, sensationally in its time, by Sparre Andersen [23], the study of the proportion J}”=1I(Sj > 0)/n of leading times has culminated in Spitzer’s [24] generalized arc-sine law: The proportion has a non-degenerate limiting distribution if and only if
(1) 1 X \
— > P{£i > 0} —> p for some p £ (0,1), n i=i
** Takacs [26] offers a masterful analysis of Levy’s heuristic idea, providing several ex
tensions and variants and delineating its limitations.
and if this condition is satisfied for p G (0,1), then
_ f 1 .. z-. / \ sin(p7r) fy dt
\ J = 1 >
(2)
r(p)r(i-p)
, o < J/ < 1, with the usual gamma function T(s) := ua 1e Sds, s > 0, that is, with the Beta(p, 1 — p) distribution function in the limit, where, for every r G N,(3)
I yrdGp(y) = Jo
T(r + p) p(p+!)•••(/>+ r-1) r(r + i)r(P)
= lim E | n—►oo
if r(S,>o) j=l
for the limiting r-th moments. The original arc-sine law is the case of p = 1/2, when
dt 2 . r- n ,
—-■ --- = — arcsin Jy, 0 < y < 1.
^(1 - V *
The latter obtains trivially if F(-) is continuous on R and F(-x) = 1 — F(x) for all x > 0, for then Sn is also symmetric about zero and so R{Sn > 0} = 1/2 for all n G N. But the case p = 1/2 occurs in (1) and (2) for asymmetric distributions as well, for example whenever E(A) = 0 and F is in the domain of attraction of the normal law, or, more generally, whenever F is in the domain of attraction of a symmetric stable law. In general, for every p G [0,1] there are families of F for which the convergence in (1) is satisfied, where the extreme cases p — 0 and p = 1 determine the corresponding asymptotically degenerate cases in (2). In fact, every P G (0,1) in (1) can be realized by families of F that belong to the domain of attraction of a suitable stable law, but, except for completely asymmetric cases, F need not be in a domain of attraction for (1) to hold. Also, what was unclear in 1956 when [24] was published, Spitzer [25] later constructed families of F for which the averages in (1) do not converge. While there are several later derivations of the generalized arc-sine law itself in the literature, by Spitzer [25] and Bingham, Goldie and Teugels [1] for instance, a most recent one is by Getoor and Sharpe [13], the analysis of Spitzer’s condition (1) is difficult and incomplete. Interesting results in this connection were given by Emery [9], Doney [6, 7] and Bingham and Hawkes [2]; for a brief discussion see pp. 379-380, 396-397 in [1]. We also note that, just as
the original theorem of Spitzer, the theorem below may be stated for the proportions 52”=1 I(Sj > 0)/n of non-negative sums, instead of those of positive sums, as well; this is because n-1 P{5j = 0} -+ 0 for any F not degenerating at zero.
Renyi-mixing of positive sums in the arc-sine law of Erdos and Kac [11]
for independent but not necessarily identically distributed random variables all having zero mean and unit variance, and the sums themselves satisfying the central limit theorem, has been established by Dzhamirzaev [8]. It is pointed out at the end of the next section that our short and easy proof is also applicable in that situation.
Due to constraints on time at this writing, the related problem of Renyi- mixing of the occupation times 1(Sj € B) for bounded Borel sets B will be considered in a subsequent note.
2. Renyi-mixing of positive sums. The main result of the present note is the following
Theorem. The sequence { > 0)/ra}^Li ,s Renyi-mixing with a non-degenerate limiting distribution if and only if Spitzer’s condition (1) holds, in which case
P<^ Gp(y)P{A}, 0<y<l,
j=l
for every event A € A and
Z^>o)
j=i
r(5 + P)
r(s + i)r(p) ’ s > 0, for every event A € A of positive probability.
Proof. Taking A = fi in the definition of Renyi-mixing for the sequence fn := n-1 52"=1 I(Sj > 0) € [0,1], the necessity of (1) follows by its neces
sity in Spitzer’s theorem. (The latter is trivial since if the bounded sequence {£„} converges in distribution, then E(fn) - n_1 Sj=i P{5j > 0} 6 [0,1]
must also converge to the mean of the limiting distribution function G, the support of which is contained in [0,1]. The limit p must then be in (0,1), otherwise G degenerates at 0 or 1.)
Conversely, suppose that (1) holds. Then, of course, P(-) cannot be degenerate. Standard estimates of the concentration function of Sn imply
that n
yP{-x < Sj < x} < Cfx , x > 0 ,
j=i
for any non-degenerate F, for some constant Cf depending on F\ this follows from Corollary 1 of Kesten [14], for example. Hence both
(4) ^P{-x<SJ<0}
n 7 = 1
and — 1 n P{0 < Sj < x}
7=1
for every x > 0, or, what is the same, (1) implies that n 1 52"=1 P{Sj >
x} —> p for each fixed x e R. Then by the Markov inequality both
i £ Ą-! < s, < 0)-L
7 = 1
and - 1 n /(0< Sj < x)Xq
7=1
p>
for every x > 0, where —> denotes convergence in probability. Hence we see by Slutsky’s elementary theorem that Spitzer’s condition (1) in fact implies that
(5) n >x)<y >
j=i
for each fixed x 6 R.
Since n-1 A ^7 > °) converges to zero in probability for each fixed k € N, by Renyi’s characterization of his mixing and by Slutsky’s theorem again, it suffices to show that
Pn,fc(y) := P-
n «
i £ nsj>«)<v n
n j=fc+i
GP(y)r T£/(S;>0)<y1 * k 7=1
5=1
> =: Pk(y), 0 < y < 1,
for each fixed k e N. Let J(x) = 1 or J(x) = 0 according as x > 0 or x < 0, consider the Borel set Bk(y) '■= {xk '•= '■ ^2j=i J(xj) S ky} C Rfc and, for typographical convenience, set also Sk := (Si,... , Sk).
Understanding vectorial inequalities to hold if and only if they hold for all respective components, using the law of total probability, the independence of the terms of the underlying sequence and the stationarity of the whole
sequence, for every y G [0,1] and k G N we have
Pn,fc(j/) = [ ^2/(Sfc + Ai+i+ --- + JQ >0)< 2/ Sk = B,(„) j=k+1
x dP{Sk < Xfc}
= y pU ^/(^+xfc+i+-..+xJ>o)<
dF{Sk < a:*}
j=k+l Bk(y)
B*(jz) J-1
x dP{5i < xi,... ,Sk < a;*}
Gp(j/)P{(S1,...,Sfc)GB,(j/)}=pfc(2/),
the convergence in the last step is by (5) and the bounded convergence theorem.
If P{4} > 0 for an A G A, then the first statement just proved means of course nothing but P{£„ < y | A} —> Gp(y), 0 < y < 1, and since, skipping the second equation, (3) holds for any s > 0 replacing r G N, the conditional moments must converge as stated. ■
Now let Yi,Y2,... be independent, but not necessarily identically dis
tributed random variables such that E(Yj) = 0 and E(Yj2) = 1 for all j G N, and that the asymptotic distribution of [Yi + • • • + Yn]/\/n is standard normal. Then the arc-sine law of Erdos and Kac [11] holds, and in a some
what lengthy and complicated proof Dzhamirzaev [8] shows that in fact the sequence {n-1 $2”=1 /(Yi + • • • +Yj > 0)} is Renyi-mixing with the limit
ing distribution function G4/2(-)- (At one point in his proof, Dzhamirzaev refers for justification to an other result of Erdos and Kac [10] for the as
ymptotic distribution of the maximum partial sums of independent and identically distributed random variables. Replacing this reference by one to Theorem 9 of Renyi [17] which extends the result in question to the non-identically distributed case, the proof appears to be correct.) Since, for any k G N, the Erdos and Kac [11] conditions above also hold for the sequence Yk+i, Yk+2,..and hence their arc-sine law is valid for the partial sums {5j(A:) := Yk+i + ••• + Yj}^.fc+1 just as well, and since asymptotic normality implies that n_1 P{-x < Sj(fc) < x} —► 0, x > 0, as an analogue of (4), we see that
n
n
E
j=k+lI(Sj(k) > x) < y - Gi/2(2/),
0 <
y< 1,
(6)
for each fixed x € R and k £ N. Using this in place of (5), the rest of the proof above works and hence Dzhamirzaev’s theorem follows.
3. Extension of Spitzer’s law for Revesz-dependent sequences. Let Q be any probability measure on the (^algebra A of the probability space (fi,X,P) such that Q<P. Then Theorems 1 and 2 of Renyi [18] and the theorem above imply that under condition (1) we also have (2) with P re
placed by Q, and hence also (3) for the expectation E<q)(-) with respect to Q, and of course Spitzer’s condition (1) is necessary for this to happen, or, what is the same, (1) must hold for Q as well. This fact offers the possibil
ity to extend the original limit theorem to sequences of weakly dependent random variables, the distributions of which are absolutely continuous with respect to the distribution of the original sequence of independent random variables.
One such situation is known to the present author: We say that a se
quence of (not necessarily identically distributed) random variables V), V2,...
is Revesz-dependent on (fi,,4,P) (or almost independent in the sense of Revesz) if there exists a sequence r\,r2,... of non-negative numbers such that for an arbitrary system of intervals (ai, hi], (a2, h2], • • •, where aj < bj and a.j,bj G {-00} URU {00}, j G N, we have
|P{n"=1 {<b < Vj < bj}} - Pin;-/ {aj < Vj < 6j}}P{an < V„ < Ml
< rnp{ n”"i {er, < Vj < 6j}}P{an <Vn< bn} for every n G N and rn < °°- Then, if V := {MM is Revesz-dependent, the proof of Theorem 2 of Revesz [21] demonstrates that the distribution Cv of the sequence V is absolutely continuous with respect to the distribution Cw of a sequence W := {of independent random variables such that P{Wj < 1} = P{Vj < x}, x G R, for all j £ N. Hence we have the following implication of the Theorem:
Corollary 1. Let Xf, Xj, • • • be identically distributed Revesz-dependent random variables with partial sums S* := + • • • + Xj, j G N. If a se
quence X\, X2,... of independent random variables with the same common marginal distribution as that of Xj, X2*,... satisfies condition (1), then (2) and (3) hold true for the sequence {Sj}j2.1 as well.
Of course, Revesz’s almost independence is a rather weak form of de
pendence. Considering how trivial is the Renyi-mixing of positive sums of independent random variables once we have (5) or (6), one cannot hon
estly hope for more from this source. However, even the slight departure from independence in Corollary 1 is perhaps of some interest if indeed it is
the first such departure; the present author is unaware of any other in the generalized arc-sine law.
If Yj*, Y2‘, • • • are n°t necessarily identically distributed Revesz-dependent random variables with zero mean and unit variance such that the corre
sponding independent sequence Yi,Y2,..., with P{Yj < a;} = P{Yj* < a;}, a: € R, j € N, satisfies the Lindeberg condition, then by Dzhamirzaev’s theorem (demonstrated above) it also follows that
n
F{n_1 £ AV + • • • + V > 0) < y} - G1/2(j/), 0 < j, < 1, j=i
as an extension of the arc-sine law of Erdos and Kac [11]. In this case, how
ever, presumably better extensions must follow directly from generalizations of Prohorov’s weak convergence theorem for the partial sum process, with the limit being the distribution of Brownian motion, to various weakly de
pendent sequences; we did not explore the literature in this connection.
Revesz [21] stated his result in the context of limit distributions for sums of random variables, which was at the time the only known example of Renyi-mixing of random variables for independent terms. Since then this mixing property has been shown to accompany theorems on limiting distri
butions for various other functions of independent random variables. Every such theorem extends to Revesz-dependent sequences. Examples are Gne
denko’s theorems for maxima and the four theorems of Erdos and Kac [10]
for maxima of partial sums and of moduli of partial sums and for sums of moduli and squares of partial sums, discussed in [4] with earlier references.
(It is a natural question in itself whether Revesz’s form of weak depen
dence for V is the best to ensure Cy <C any form would work with Renyi-mixing.)
4. The generalized arc-sine law for a random number of games.
Preserving the notation of the first two sections, let additionally
and {/zn}£Lj be two sequences of random variables, which may be chosen to be the same and are given on the same probability space (fi, A, P) where the basic sequence {XJnA of independent and identically distributed variables is defined, such that and fj,n for each n € N take on only positive integers as possible values and, for the same non-decreasing sequence of positive numbers such that dn —> oo, both
and
for some random variables v and /z for which P{p > 0} = 1 and P{/z > 0} = 1.
A special case of Theorem 1 in [4] says that if the sequence of ratios of the terms of an almost surely non-decreasing sequence of random variables to the corresponding terms of a regularly varying norming sequence of positive constants is Renyi-mixing with a non-degenerate limiting distribution func
tion <?(•), as n —> oo, then so is the randomly selected subsequence of ratios indexed by un as above. Very recently Kruglov and Zhang [15] proved that (for a sequence {vn} being fixed) the converse is also true, at least when the regularly varying norming sequence has a positive exponent and is non
decreasing itself: Renyi-mixing of the randomly indexed sequence of ratios with a limiting distribution function G(-) implies the same for the original sequence of ratios.
Putting together the theorem above and these results, along with a con
sequence of Renyi-mixing for the randomly indexed sequence specialized from Theorem 2 in [4], we obtain
Corollary 2. The sequence { I(Sj > Oj/Pn}^ is Renyi-rnixing with a non-degenerate limiting distribution if and only if Spitzer’s condition (1) holds, in which case
-i- £
r(Sj > 0) < yj=l
for every event A € A, where Gp(t) = 1 for t > 1, and
(7) EKSj>0)
Lj=i
r(« + p) r(s + i)r(p)’
for every event A £ A of positive probability.
In particular, to spell out some unconditional special cases, if (1) holds, then
Under condition (1), this statement contains of course that (8) 4-i>(Ą>0)<j
• -+ Gp(y), 0 < y < 1, J=1
which is the essence of Corollary 2, and the entertaining variants
-J-
Y'
I(Sj > 0) < y > — P{j/ < y} + [ Gp(-\dP{v<x}, y>Q,dn f Jy XX/
n A
E s > 0,
and
, LdnJ
- E
HSj > 0) < J, /in j=lGp(ij/)dP{^ < a;} , y > 0 .
More interesting than these last two limit statements is to close with some concrete examples for (8), in which {z/n} is constructed from the sequence Xi,X2,-- - itself. Besides (1), suppose also that := E(|A"|7) 6 (0,oo) for some 7 > 0. (For example, if 7 > 2, then (1) holds with p — 1/2. On the other hand, if F is in the domain of attraction of a stable law with characteristic exponent a G (0,2], skewness parameter (3 G ( — 1,1) and zero location parameter, where we take (3 = 0 if a = 1 and assume that E(X) = 0 if a G (1,2], then (1) is satisfied with p = 1/2 for a — 1 and with p = I + arctan (/? tan ^) for a / 1, and any 7 € (0, a) works; see [5] for instance concerning the details here.) Let 72/ := |X\ |7 H--- (-1Xfc|7, k G N, and for each n £ N, define i/n(l) := 72/, p„(2) := min{A: G N : 72/ > n} , i/n(3) := max{/c G N : 72/ < n} and pn(4) := min{fc G N : 72/ > nk6}
for some 6 G (0,1). Then the strong law of large numbers implies that j/n(l)/ra —> , i/„(2)/n —> m"1 2 3 , p„(3)/n->■ m/4 5 6 7 8 9 and pn(4)/n1/(1-'5) _>
^i/(<5—1), jn each case aimost surely; see Section 5.4 in [3] for example. Thus we have (7) and (8) for all four choices of vn = I = 1,2,3,4.
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Department of Statistics received February 5, 1997 The University of Michigan
Ann Arbor, MI 48109-1027, U.S.A.
Bolyai Institute University of Szeged Aradi vertanuk tere 1 H-6720 Szeged, Hungary
